I would like to add one additional recommendation that I think would benefit any drink with egg in it:
Use chopsticks or tweezers to remove the white bouncy knot-of-umbilical-chord that is attached to the yoke (the technical term for that thing is the chalaza) before blending or shaking the drink. It sometimes comes loose when separating the egg, and no matter how much you shake or blend, it never incorporates well into the drink. The typical hawthorne strainer/coil strainer does not strain out the chalaza from an already mixed drink, and it certainly is not pleasant to accidentally sip a chunk of chalaza into your mouth when trying to enjoy a cocktail.

@blair frodelius:
Virtually all of the difference between organic and conventional eggs is in the yoke, not the egg white. Although I have not tested this myself, as far as I understand, the egg yoke is identical between organic and non-organic eggs.
Also, if you're making a drink with a whole egg, as I mentioned before, something I recommend is to remove the chalaza from the egg, since no amount of shaking or blending will incorporate it into the drink.

Jaime,
I know that eggs of differing freshness and grade have differing egg white and yoke firmness, with the highest grade and freshest eggs giving the firmest egg parts. For example, eggs served sunny side up ought to be AA and very fresh, whereas the difference between AA, A, and B is negligible when the eggs are used in pancakes or scrambled. How does this impact the taste and mouthfeel of a drink? Do softer egg whites incorporate or foam better than firm egg whites? It would seem to me that a very firm fresh AA egg would not incorporate as well nor make as smooth a drink as a grade B egg or one whose egg white has softened for a few days, but I have not tested this myself.

Let me correct my prior post about the difference between organic and non-organic eggs: I meant that the *egg white* is identical between organic and non-organic eggs, not the yoke. Organic egg yokes usually look and taste different from non-organic egg yokes.

@Garretto:
Believe it or not, the egg came first. Chickens were domesticated from wild grouse, which have lain eggs long before chickens existed . Thus, I can say with confidence that the egg came first. ^_~

The ice shown in the video was not molded in the Taisin brass ice mold; it was most likely cast in a spherical ice mold such as this:
http://www.momastore.org/museum/moma/ProductDisplay_Spherical Ice Tray Set_10451_10001_57253_-1_11515_11521_null_shop_
At $16, it's a heck of a lot cheaper than the $170 ice mold in that Youtube video. This ice mold is also from Japan.
The device shown in the video takes the spherical ice concept (which is already an example of Japanese obsession with optimization) and takes it to another level: the Taisin ice mold is intended to melt-mold blocks of crystal clear ice into spheres with absolutely no visible seams by pressing a block of ice between two brass anvils. A "conventional" spherical ice mold will leave seams on your ice ball and a little sprue on top where the water was poured in, and is difficult to make totally clear ice in.
If you pour your water into a mold for spherical ice, it is going to be cloudy unless you use de-mineralized water that has had all the dissolved gasses boiled out. The only other way to get crystal clear ice is to freeze the water one thin layer at a time; the freezing process drives out the gasses, and as long as each layer is thin enough, the gasses escape upwards rather than getting trapped in the ice. This layered method is how some ice machines manage to produce crystal clear ice.

@Raven
The practice of spherical ice arose in Japan for drinking whiskey on the rocks very slowly (usually accompanying length conversations). Their bartenders were concerned that the drink would not be as enjoyable if the ice melted too quickly. Since spheres have the least surface area for the amount of volume they contain, they will chill a drink for the longest time with the least dilution.
The preferred drinks for use with spherical ice are hard liquors that would otherwise be served with cubed ice; in Japan, they usually serve Scotch or Japanese whiskey over spherical ice.

Hello Jaime,
You explained that one should stir the drink when using clear ingredients, but you didn't explain why stirring keeps the drink clear. I understand that shaking incorporates more air, but the air should bubble out after shaking, and any air that absorbs into the drink (akin to carbonation) shouldn't affect the transparency of the drink any more than carbonating a drink affects transparency--which is not at all. Could you do a side-by-side demo using the same ingredients to show the effectual difference between stirring and shaking?
Also, how can you tell when a drink has diluted enough? I've seen folks stir the Old-Fashioned for 3-5 minutes while adding ice claiming it needed to dilute enough, and I've seen others stir briefly, claiming that alcohol lowers the melting point of the ice and dilutes pretty rapidly to begin with. How can you tell how long is long enough without sampling the drink all the time the way Chris Doig (from brilliantcocktails.com) does?

A follow-up on my previous comment:
I pondered this problem a bit, and I think I figured out a potential answer to my own question about how shaking might cause a drink to cloud up. The air won't bubble out quickly enough to pour clearly if the drink has any viscosity because shaking would form ultra fine bubbles that remain suspended and slowly bubble out over an hour or so, while forming foam in some cases.
None the less, a video side-by-side demo to show the difference between stirring and shaking would be nice.